Problem: $ 0.\overline{91} \div 3.\overline{20} = {?} $
Answer: First convert the repeating decimals to fractions. $\begin{align*} 100x &= 91.9191...\\ x &= 0.9191...\end{align*} $ $\begin{align*} 99x &= 91 \\ x &= \dfrac{91}{99}\end{align*} $ $\begin{align*} 100y &= 320.202...\\ y &= 3.202...\end{align*} $ $\begin{align*} 99y &= 317 \\ y &= \dfrac{317}{99}\end{align*} $ So, the problem becomes: $ \dfrac{91}{99} \div \dfrac{317}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{91}{99} \times \dfrac{99}{317} = {?} $ $ \phantom{\dfrac{91}{99} \times \dfrac{317}{99}} = \dfrac{91 \times 99}{99 \times 317} $ $ \phantom{\dfrac{91}{99} \times \dfrac{317}{99}} = \dfrac{91 \times \cancel{99}} {\cancel{99} \times 317} $ $ \phantom{\dfrac{91}{99} \times \dfrac{317}{99}} = \dfrac{91}{317} $